Hyperbolic Solids in Blender?
If you're just in it for the looks, the easiest thing would be to take advantage of proportional editing. Using your cube as an example, do the following steps:


Sub-divide the cube multiple (Edit mode > Select All [A] > Specials [W] > Subdivide... then use the Last Operator panel [F6] to increase the number of subdivisions to taste).
Select just the eight corner vertices on the cube.
Enable proportional editing (O).
Change the proportional fall-off to Sharp (Shift+O 3 times if you're at the default of Smooth. Alternatively, there's a drop-down menu in the 3D View's header where you can choose the fall-off using your mouse).
Scale (S) those vertices. As you do, scroll your mouse wheel to adjust the fall-off influence of the proportional edit area.
To get sharp edges, select all the edges around the model and then crease them with (ShiftE) and drag until the value in the header is at 1.0 or just press 1 on your keyboard.


Finally, you may want to also use the Subdivision Surface modifier and the Edge Split modifier to help smooth things out a bit.